Exact ground state Monte Carlo method for Bosons without importance sampling

Abstract

Generally "exact" Quantum Monte Carlo computations for the ground state of many Bosons make use of importance sampling. The importance sampling is based, either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of Path Integral Ground State (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a non zero overlap with the ground state, that gives rise to a discrete path which is sampled via a Metropolis like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid 4 He with PIGS by considering as initial wave function a constant, i.e. the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e. there is no importance sampling. For both the phases we obtain results converging to those obtained by considering the best available variational wave function (the Shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature Path Integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.

Simulation and understanding of atomic and molecular quantum crystals

“…II.B.1). Recent applications of the PIGS method involving high-order decomposition methods have shown that it is actually possible to obtain results that are completely independent of the trial wave function that is used as boundary condition (Rossi et al, 2009;Rota et al, 2010). Even in the limiting case of considering only the symmetry requirement of the system (e. g., ψ(R) = 1 in the bosonic case) the PIGS method works reliably, with the only penalty of producing slightly larger variances.…”

Observation of crystallization slowdown in supercooled parahydrogen and orthodeuterium quantum liquid mixtures

“…To access static structural properties of the nonequilibrium pH 2 -oD 2 quantum liquid mixtures we have carried out path-integral Monte Carlo (PIMC) simulations [29] by using a canonical [30] Worm algorithm [31]. We have simulated mixtures of up to 300 molecules in boxes with periodic boundary conditions with 0, 3, and 10% oD 2 at T = 13 K, with 50% oD 2 at T = 14.5 K, and with 90, 97, and 100% oD 2 at T = 17 K. In order to avoid the crystallization of the mixtures we followed the strategy reported in Ref.…”

Roton Excitations and the Fluid–Solid Phase Transition in Superfluid 2D Yukawa Bosons

“…The PIGS method provides estimates of ground-state properties and imaginary-time correlations functions, which are only affected by two errors: (i) the use of a finite imaginary time of projection in (8), and (ii) the use of a finite time step in (9). The biases introduced by these approximations can be reduced below the statistical uncertainties of the calculation by taking τ sufficiently large and δτ sufficiently small [43]. For the small imaginary-time propagator we have used the pair-product approximation [4] with imaginary time step δτ = 120ǫ −1 at ρ * = 0.01, δτ = 60ǫ −1 at ρ * = 0.0175 and δτ = 30ǫ −1 at ρ * = 0.0225; total imaginary projection time, τ , ranges from 2400 ǫ −1 to 4800 ǫ −1 depending on the density.…”

“…The quality of the trial wavefunction, however, has a determining impact on the efficiency and accuracy of the calculation [43]. Shadow wavefunctions [19,[32][33][34][35] (SWFs) take into account interparticle correlations by introducing auxiliary variables S = (s 1 .…”